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Abstract Lymphatic vessel dysfunction is increasingly recognized in metabolic diseases, contributing to edema, dyslipidemia, and tissue lipid accumulation. Type 2 diabetes has been experimentally linked to lymphatic vascular defects characterized by enhanced permeability resulting from low nitric oxide (NO) bioavailability. To examine the effects of this dysfunction, we developed a mathematical model of fluid exchange between a collecting lymphatic vessel and the surrounding tissue. The lymph flow is modeled using the Stokes equations for an incompressible, viscous fluid, while the collecting vessel is represented by a thin Koiter shell model. The main novelty of our approach is that the interstitium is described as poroelastic, and modeled using the Biot equations, which capture both porous medium flow and the displacement. Lymph movement within the vessel is regulated by biological factors, specifically calcium ion ($$\text {Ca}^{2+}$$ ${\text{Ca}}^{2+}$ ) concentration and NO production, which are included to modulate the pumping mechanism. Using this model, we quantify lymph flow and fluid exchange for both wild-type (WT) and diabetic vessels. Our findings illustrate how variations in NO production rate, Young’s modulus, and hydraulic conductivity impact leakage and pumping efficiency. The results demonstrate that reduced NO production in diabetic conditions leads to increased fluid leakage and altered pulsation frequency, while increased vessel stiffness further compromises lymphatic function. 

Abstract Efficient and accurate numerical solvers for fluid-poroelastic structure interaction problems are critical to a variety of applications in geomechanics, biomedical engineering, and subsurface flow modeling. This work introduces and analyzes a partitioned method based on Robin-Robin coupling conditions for fluid-poroelastic structure interaction. We describe the fluid subproblem using the time dependent Stokes equations, and the poroelasticity subproblem using the Biot model. We propose to solve the fluid subproblem separately from the Biot subproblem, and to further decompose the Biot subproblem into a mechanics subproblem and a Darcy subproblem. We first consider a numerical method based on the Backward Euler time discretization, and show that it is conditionally stable, and sub-optimally convergent, with the rate of$$\mathcal {O}(\tau ^\frac{1}{2})$$ $O\left({\tau }^{\frac{1}{2}}\right)$ . To improve the rate of convergence, we introduce a second numerical method based on Cauchy’s ‘$$\theta $$ $\theta$ -like’ scheme combined with Robin-Robin coupling conditions. This method consists of an iterative Backward Euler step followed by a Forward Euler step which is equivalent to a linear extrapolation. The iterative procedure is proved to be convergent assuming certain conditions on the time step and material parameters, under which the converged solution has also been shown to be stable. We further investigate the convergence rates and the impact of the Robin-Robin coupling parameter,L, using numerical examples. These examples show that the sub-optimal convergence of the first method is only observed whenLis small. They also show that the iterative procedure decreases errors and increases the rates of convergence. 

Abstract This work focuses on the development of a novel, strongly-coupled, second-order partitioned method for fluid–poroelastic structure interaction. The flow is assumed to be viscous and incompressible, and the poroelastic material is described using the Biot model. To solve this problem, a numerical method is proposed, based on Robin interface conditions combined with the refactorization of the Cauchy’s one-legged ‘ϑ-like’ method. This approach allows the use of the mixed formulation for the Biot model. The proposed algorithm consists of solving a sequence of Backward Euler–Forward Euler steps. In the Backward Euler step, the fluid and poroelastic structure problems are solved iteratively until convergence. Then, the Forward Euler problems are solved using equivalent linear extrapolations. We prove that the iterative procedure in the Backward Euler step is convergent, and that the converged method is stable whenϑ∈ [1/2, 1]. Numerical examples are used to explore convergence rates with varying parameters used in our scheme, and to compare our method to a monolithic method based on Nitsche’s coupling approach. 

Abstract We consider the interaction between a poroelastic structure, described using the Biot model in primal form, and a free-flowing fluid, modelled with the time-dependent incompressible Stokes equations. We propose a diffuse interface model in which a phase field function is used to write each integral in the weak formulation of the coupled problem on the entire domain containing both the Stokes and Biot regions. The phase field function continuously transitions from one to zero over a diffuse region of width $$\mathcal{O}(\varepsilon)$$ around the interface; this allows the weak forms to be integrated uniformly across the domain, and obviates tracking the subdomains or the interface between them. We prove convergence in weighted norms of a finite element discretization of the diffuse interface model to the continuous diffuse model; here the weight is a power of the distance to the diffuse interface. We, in turn, prove convergence of the continuous diffuse model to the standard, sharp interface, model. Numerical examples verify the proven error estimates, and illustrate application of the method to fluid flow through a complex network, describing blood circulation in the circle of Willis. 

We develop a computational algorithm based on a diffuse interface approach to study the design of bioartificial organ scaffold architectures. These scaffolds, composed of poroelastic hydrogels housing transplanted cells, are linked to the patient’s blood circulation via an anastomosis graft. Before entering the scaffold, the blood flow passes through a filter, and the resulting filtered blood plasma transports oxygen and nutrients to sustain the viability of transplanted cells over the long term. A key issue in maintaining cell viability is the design of ultrafiltrate channels within the hydrogel scaffold to facilitate advection-enhanced oxygen supply ensuring oxygen levels remain above a critical threshold to prevent hypoxia. In this manuscript, we develop a computational algorithm to analyze the plasma flow and oxygen concentration within hydrogels featuring various channel geometries. Our objective is to identify the optimal hydrogel channel architecture that sustains oxygen concentration throughout the scaffold above the critical hypoxic threshold. The computational algorithm we introduce here employs a diffuse interface approach to solve a multi-physics problem. The corresponding model couples the time-dependent Stokes equations, governing blood plasma flow through the channel network, with the time-dependent Biot equations, characterizing Darcy velocity, pressure, and displacement within the poroelastic hydrogel containing the transplanted cells. Subsequently, the calculated plasma velocity is utilized to determine oxygen concentration within the scaffold using a diffuse interface advection-reaction-diffusion model. Our investigation yields a scaffold architecture featuring a hexagonal network geometry that meets the desired oxygen concentration criteria. Unlike classical sharp interface approaches, the diffuse interface approach we employ is particularly adept at addressing problems with intricate interface geometries, such as those encountered in bioartificial organ scaffold design. This study is significant because recent developments in hydrogel fabrication make it now possible to control hydrogel rheology and utilize computational results to generate optimized scaffold architectures. 

This work is focused on the mathematical and computational modeling of bioconvection, which describes the mixing of fluid and micro-organisms exhibiting negative geotaxis movement under the force of gravity. The collective population moves towards the surface of the fluid, generating a Rayleigh–Taylor instability, where initial fingers of organisms plummet to the bottom. The inherent drive to swim vertically generates large collective flow patterns that persist in time. We model the flow using the Navier–Stokes equations for an incompressible, viscous fluid, coupled with the transport equation describing the concentration of the micro-organisms. We use a nonlinear semigroup approach to prove the existence of solutions. We propose a partitioned, second-order, time adaptive numerical method based on the Cauchy’s one-legged ‘-like’ scheme. We prove that the method is energy-stable, and for small time steps, the iterative procedure in the partitioned algorithm is linearly convergent. The numerical results confirm the expected second-order of accuracy. We also present a computational study of a chaotic system describing bioconvection of motile flagellates. 

We consider the interaction between a free flowing fluid and a porous medium flow, where the free flowing fluid is described using the time dependent Stokes equations, and the porous medium flow is described using Darcy’s law in the primal formulation. To solve this problem numerically, we use a diffuse interface approach, where the weak form of the coupled problem is written on an extended domain which contains both Stokes and Darcy regions. This is achieved using a phase-field function which equals one in the Stokes region and zero in the Darcy region, and smoothly transitions between these two values on a diffuse region of width (ϵ) around the Stokes-Darcy interface. We prove convergence of the diffuse interface formulation to the standard, sharp interface formulation, and derive rates of convergence. This is performed by deriving a priori error estimates for discretizations of the diffuse interface method, and by analyzing the modeling error of the diffuse interface approach at the continuous level. The convergence rates are also shown computationally in a numerical example.